ICM ~Unit 4 ~ Day 1

Section 1.2—Domain, x & y intercepts

Warm Up ~ What do you remember???

For #1 and #2, for each question:

a. Factor, if possible

b. Solve

Simplify Completely1

2 3 23 9. ( )x y

21 25 0 . x22 17 0 . x

24 5 3 2 . ( ) ( )f x given f x x x

Discuss HW

• Factoring & Function Review

What is a Function?

• A rule that associates every value in a set of numbers (domain) with a SINGLE value in another set of numbers (range).

• Notation: f(x) is the value of the function at a given value of “x”

Function vs Not Function• Vertical line test—used to graphically verify

that the domain values are associated with only ONE range value

Definition of Domain:• Set of real numbers that result in a “defined”

or mathematically possible statement.

• Use interval notation! ( ) Not included [ ] included

• Examples:

(0, 10]

• Never write ones like

( , ) ( ,0] [1, )

(0,-¥) Smallest value must always be on the left!

Real Life Examples

• Domain for ages of students at GHHSD: [13, 21] agesR: [1, 500] number of students per grade

• What is the domain for a cash payment of $6.47? (what are all possible ways to pay this amount?)

D: pennies, nickels, dimes,

quarters, $1, $5, dollar coin, $2

R: amount of each coin, [0, 647]

X and Y intercepts – Special Points!• X-intercepts (X, 0)

– Where the function crosses the x-axis

– Zero or root of the function*Set y = 0 and solve for x

Ex 1.

Ex 2.

2 5

2

xy

225

0 , 5, 52

xx x

( 5,0) ( 5,0)and

24 18 20y x x 5

(2,0) ( ,0)2

and

X and Y intercepts – Special Points!• Y-intercepts (0, Y)

– Where the function crosses the y-axis

– Where x equals 0

*Set x = 0 and solve for y

Ex 1.

Ex. 2

20 5 5,

2 2y y

5(0, )

2

(0,20)

2 5

2

xy

24 18 20y x x

State the Domain and x & y int:

2( )

( )

1( )

f x x

f x x

f xx

Domain : (-¥,¥)

x- int : (0,0)

y - int : (0,0)

Domain : [0,¥)

x - int : (0,0)

y - int : (0,0)

: ( ,0) (0, )

int :

int :

Domain

x none

y none

Radical Examples From Class

5: [ , )

3

( ) 3 5

Dom

x

a n

f

i

x

2( ) 3 11 4

1: ( , 4] [ , )

3Doma

x x

in

g x

4 4:( , )

3( )

5 4

( , )5 5

Domain

xf x

x

Rational Examples From Class

2

3 3:( , 7) ( 7, ) ( , )

2

2 3( )

2 11 21

2Domain

xg x

x x

Square Root Functions

Steps to find domain of Square Root Functions:

1. Set the expression inside the square root equal to zero.

2. Solve the equation found in step 1.

3. Plot that value on a number line and test points on each side.

4. Write the domain using interval notation.

( )f x radicand

( ) 9 5f x x9

(: , ]5

Domain

9 5 0

9 5

9

5

x

x

x

Find the x and y-intercepts.9

( , 0)int : int : (0, 3)5

x y

Expression under the square root symbol, must be positive.

*Write this

down!

Example: Find the domain.

Square Root Functions

• Write the domain, using interval notation, of the following functions. (verify by graphing)

• State the x & y intercepts

( ) 2 7f x x

:

int :

7[ , )

2

7( , 0)

2

(0, 7)int :

Domain

x

y

Domain : (-¥, -3] È [3, ¥)

x- int : (-3, 0), (3, 0)

y - int : None

2( ) 9h x x

( )f x radicand

You Try!

Rational Functions ( )min

numeratorf x

deno ator

2

1( )

2 35

xf x

x x

( , 7) ( 7,5) (5, )

Steps to find domain of Rational Functions:

1. Factor the denominator (if possible).

2. Set each factor equal to zero and solve.

3. Write the domain using interval notation.

Expression in the denominator can NOT be equal to zero.

Find the x and y-intercepts.

Domain: ( 1, 0)int :

int : (0, 1 35)

x

y

*Write

this

down!

Rational FunctionsYou Try!

( )min

numeratorf x

deno ator

2

2

8. ( ) . ( ) 8 15

2 12

7. ( )

16

a g x b f x x xx

c k xx

Domain : (-¥,-6)È (-6,¥)

x - int : None

y - int : (0,2

3)

( , 4) ( 4,4) (4:

in

,

t :

int :

)

(0, 7 16)

Domain

x e

y

Non

:

int :

( ,3] [5, )

(3,0), (5, 0)

(0, 15)int :

Domain

x

y

• Write the domain, using interval notation, of the following functions. (verify by graphing)

• State the x & y intercepts

Combined FunctionsTo find the domain of combined functions, a number line really helps to put the pieces together!!

• Write the domain of the following function in interval notation (verify by graphing)

• State the x & y intercepts

( )9

xm x

x

Domain : [0, 9)È (9,¥)

x- int : (0, 0)

y - int : (0, 0)

Practice • Find the domain and x & y intercepts of…

1 5

3f x

x x

24

3

xh x

x

2

4

1 1

xg x

x x

( ,0) (0,3) (3, ):

int : (0.5, 0)

int :

Do

None

main

x

y

Domain : [-2, 2]

x - int : (2, 0), (-2, 0)

y - int : (0, -2

3)

( , 1) ( 1,4]

(4, 0)

(0, 2)

:

int :

int :

Domain

x

y

• Packet p. 1 Homework Day 1